Laura Bartoli , Simone Cacace , Emiliano Cristiani , Roberto Ferretti
{"title":"可变最大密度的宏观行人模型","authors":"Laura Bartoli , Simone Cacace , Emiliano Cristiani , Roberto Ferretti","doi":"10.1016/j.amc.2025.129404","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper we propose a novel macroscopic (fluid dynamics) model for describing pedestrian flow in low and high density regimes. The model is characterized by the fact that the maximal density reachable by the crowd – usually a fixed model parameter – is instead a state variable. To do that, the model couples a conservation law, devised as usual for tracking the evolution of the crowd density, with a Burgers-like PDE with a nonlocal term describing the evolution of the maximal density. The variable maximal density is used here to describe the effects of the psychological/physical pushing forces which are observed in crowds during competitive or emergency situations.</div><div>Specific attention is also dedicated to the fundamental diagram, i.e., the function which expresses the relationship between crowd density and flux. Although the model needs a well defined fundamental diagram as known input parameter, it is not evident <em>a priori</em> which relationship between density and flux will be actually observed, due to the time-varying maximal density. An <em>a posteriori</em> analysis shows that the observed fundamental diagram has an elongated “tail” in the congested region, thus resulting similar to the concave/concave fundamental diagram with a “double hump” observed in real crowds.</div><div>The main features of the model are investigated through 1D and 2D numerical simulations. The numerical code for the 1D simulation is freely available <span><span>on this Gitlab repository</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"499 ","pages":"Article 129404"},"PeriodicalIF":3.5000,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A macroscopic pedestrian model with variable maximal density\",\"authors\":\"Laura Bartoli , Simone Cacace , Emiliano Cristiani , Roberto Ferretti\",\"doi\":\"10.1016/j.amc.2025.129404\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this paper we propose a novel macroscopic (fluid dynamics) model for describing pedestrian flow in low and high density regimes. The model is characterized by the fact that the maximal density reachable by the crowd – usually a fixed model parameter – is instead a state variable. To do that, the model couples a conservation law, devised as usual for tracking the evolution of the crowd density, with a Burgers-like PDE with a nonlocal term describing the evolution of the maximal density. The variable maximal density is used here to describe the effects of the psychological/physical pushing forces which are observed in crowds during competitive or emergency situations.</div><div>Specific attention is also dedicated to the fundamental diagram, i.e., the function which expresses the relationship between crowd density and flux. Although the model needs a well defined fundamental diagram as known input parameter, it is not evident <em>a priori</em> which relationship between density and flux will be actually observed, due to the time-varying maximal density. An <em>a posteriori</em> analysis shows that the observed fundamental diagram has an elongated “tail” in the congested region, thus resulting similar to the concave/concave fundamental diagram with a “double hump” observed in real crowds.</div><div>The main features of the model are investigated through 1D and 2D numerical simulations. 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A macroscopic pedestrian model with variable maximal density
In this paper we propose a novel macroscopic (fluid dynamics) model for describing pedestrian flow in low and high density regimes. The model is characterized by the fact that the maximal density reachable by the crowd – usually a fixed model parameter – is instead a state variable. To do that, the model couples a conservation law, devised as usual for tracking the evolution of the crowd density, with a Burgers-like PDE with a nonlocal term describing the evolution of the maximal density. The variable maximal density is used here to describe the effects of the psychological/physical pushing forces which are observed in crowds during competitive or emergency situations.
Specific attention is also dedicated to the fundamental diagram, i.e., the function which expresses the relationship between crowd density and flux. Although the model needs a well defined fundamental diagram as known input parameter, it is not evident a priori which relationship between density and flux will be actually observed, due to the time-varying maximal density. An a posteriori analysis shows that the observed fundamental diagram has an elongated “tail” in the congested region, thus resulting similar to the concave/concave fundamental diagram with a “double hump” observed in real crowds.
The main features of the model are investigated through 1D and 2D numerical simulations. The numerical code for the 1D simulation is freely available on this Gitlab repository.
期刊介绍:
Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results.
In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.